Question:

The rate of a first order reaction is $0.04 mol \, l^{-1} \, s^{-1}$ at 10 seconds and $0.03 \, mol \, l^{-1} \, s^{-1}$ at 20 seconds after initiation of the reaction. The half-life period of the reaction is :

Updated On: Apr 20, 2025
  • 34.1 s
  • 44.1 s
  • 54.1 s
  • 24.1 s
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The Correct Option is D

Solution and Explanation

Calculation of Half-Life 

We are given the equation for the rate constant K as:

\(K = \frac{2.303}{10} \log \frac{0.04}{0.03}\)

Now, simplifying the logarithmic expression:

\(K = \frac{2.303}{10} \log \frac{4}{3}\)

Using the logarithmic value:

\(K = \frac{2.303 \times 0.123}{10}\)

Now, calculate K:

\(K = 0.0285\)

The half-life t1/2 is calculated using the formula:

\(t_{1/2} = \frac{0.693}{K}\)

Substitute the value of K into the equation:

\(t_{1/2} = \frac{0.693}{0.0285} = 24.1 \, \text{s}\)

Thus, the half-life is 24.1 seconds.

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Concepts Used:

Chemical Kinetics

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Factors Affecting The Reaction Rate:

  • The concentration of Reactants - According to collision theory, which is discussed later, reactant molecules collide with each other to form products.
  • Nature of the Reactants - The reaction rate also depends on the types of substances that are reacting.
  • Physical State of Reactants - The physical state of a reactant whether it is solid, liquid, or gas can greatly affect the rate of change.
  • Surface Area of Reactants - When two or more reactants are in the same phase of fluid, their particles collide more often than when either or both are in the solid phase or when they are in a heterogeneous mixture. In a heterogeneous medium, the collision between the particles occurs at an interface between phases. Compared to the homogeneous case, the number of collisions between reactants per unit time is significantly reduced, and so is the reaction rate.
  • Temperature - If the temperature is increased, the number of collisions between reactant molecules per second. Increases, thereby increasing the rate of the reaction.
  • Effect Of Solvent - The nature of the solvent also depends on the reaction rate of the solute particles.
  • Catalyst - Catalysts alter the rate of the reaction by changing the reaction mechanism.